Shannon information entropies for position-dependent mass Schrödinger problem with a hyperbolic well

The Shannon information entropy for the Schrödinger equation with a nonuniform solitonic mass is evaluated for a hyperbolic-type potential. The number of nodes of the wave functions in the transformed space z are broken when recovered to original space x. The position S_x and momentum S_p information entropies for six low-lying states are calculated. We notice that the S_x decreases with the increasing mass barrier width a and becomes negative beyond a particular width a, while the S_p first increases with a and then decreases with it. The negative S_x exists for the probability densities that are highly localized. We find that the probability density ρ(x) for n = 1, 3, 5 are greater than 1 at position x = 0. Some interesting features of the information entropy densities ρ_s(x) and ρ_s(p) are demonstrated. The Bialynicki-Birula-Mycielski (BBM) inequality is also tested for these states and found to hold.